Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations

We consider operator semigroups generated by Feller processes killed upon leaving a given domain. These semigroups correspond to Cauchy–Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation leads to a representation of the solution of the corresponding Cauchy–Dirichlet type problem by a Feynman formula, i.e. by a limit of n-fold iterated integrals of certain functions as n → ∞. Feynman formulae can be used for direct calculations, modelling of underlying dynamics, simulation of underlying stochastic processes. Further, a method to approximate solutions of time-fractional evolution equations is suggested. The method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. This method leads to Feynman formulae for solutions of time-fractional evolution equations. A class of distributed order time-fractional equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy–Dirichlet type problems are obtained.

Chernoff approximation of subordinate semigroups

This note is devoted to the approximation of evolution semigroups generated by some Markov processes and hence to the approximation of transition probabilities of these processes. The considered semigroups correspond to processes obtained by subordination (i.e. by a time-change) of some original (parent) Markov processes with respect to some subordinators, i.e. Lévy processes with a.s. increasing paths (they play the role of the new time). If the semigroup, corresponding to a parent Markov process, is not known explicitly then neither the subordinate semigroup, nor even the generator of the subordinate semigroup are known explicitly too. In this note, some (Chernoff) approximations are constructed for subordinate semigroups (in the case when subordinators have either known transitional probabilities, or known and bounded Lévy measure) under the condition that the parent semigroups are not known but are already Chernoff-approximated. As it has been shown in the recent literature, this condition is fulfilled for several important classes of Markov processes. This fact allows, in particular, to use the constructed Chernoff approximations of subordinate semigroups, in order to approximate semigroups corresponding to subordination of Feller processes and (Feller type) diffusions in Euclidean spaces, star graphs and Riemannian manifolds. Such approximations can be used for direct calculations and simulation of stochastic processes. The method of Chernoff approximation is based on the Chernoff theorem and can be interpreted also as a construction of Markov chains approximating a given Markov process and as the numerical path integration method of solving the corresponding PDE/SDE.

The Proof of the Feynman–Kac Formula for Heat Equation on a Compact Riemannian Manifold

A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.